The field of the invention is systems and methods for magnetic resonance imaging (“MRI”). More particularly, the invention relates to systems and methods for calculating and selecting a k-space sampling pattern or diffusion gradient table.
The problem of constructing a set of uniformly-distributed points on the surface of a sphere has a long and interesting history, which dates back to J. J. Thomson in 1904, as described by J. J. Thomson in “On the Structure of the Atom: An Investigation of the Stability and Periods of Oscillation of a Number of Corpuscles Arranged at Equal Intervals Around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure,” Philosophical Magazine, 1904; 7(39):237-265. A particular variant of the Thomson problem that is of great importance to biomedical imaging is the problem of generating a nearly uniform distribution of points on the sphere via a deterministic scheme. Although the point set generated through the minimization of electrostatic potential based Coulomb's law is the gold standard, minimizing the electrostatic potential of one thousand or more points, or charges, remains a formidable task.
Therefore, a deterministic scheme capable of efficiently and accurately generating a set of uniformly-distributed points on the sphere has an important role to play in many biomedical and engineering applications, such as three-dimensional projection reconstruction of medical images, three-dimensional selective radiofrequency pulse design in MRI, and diffusion-weighting direction design and selection in diffusion MRI. Many deterministic schemes have been proposed in the past, including those described by E. Saff and A. Kuijlaars in “Distributing Many Points on a Sphere,” The Mathematical Intelligencer, 1997; 19:5-11; by E. Rakhmanov, et al., in “Minimal Discrete Energy on the Sphere,” Mathematical Research Letters, 1994; 1:647-662; and by R. Ahmad, et al., in “Quasi Monte Carlo-based isotropic distribution of gradient directions for improved reconstruction quality of 3D EPR imaging,” Journal of Magnetic Resonance, 2007; 184(2):236-245.
The challenge remains, however, to provide a deterministic method for producing a uniform distribution of points on the surface of a sphere that is advantageous for medical imaging applications, such as those referred to above. Notably, for determining diffusion-weighting directions, the point set should present antipodal symmetry, which currently existing methods do not provide.
The importance and the effects of view-ordering on image quality in MRI has been studied extensively in many subfields, such as two-dimensional Cartesian acquisitions; radial fast spin echo (“FSE”) acquisitions; and four-dimensional MR angiography with three-dimensional radial acquisitions. For example, the basic idea of FSE is to acquire multiple echoes within each excitation or within the same repetition time (“TR”). FSE acquisitions can provide for a significant reduction in scan time, which can be used to improve image resolution. However, the acquisition of multiple echoes within the same excitation comes at a cost of enhanced image artifacts, such as blurring or ringing because of T2 decay, which produces signal modulations in k-space. Therefore, different view-ordering strategies have been developed and used in an effort to make the signal modulation as incoherent as possible in k-space.
Among the many studies on the relative merits of various diffusion gradient schemes, it is well accepted that the uniformity of the diffusion gradient schemes plays an important role in the final estimate of any diffusion MRI or diffusion tensor-derived quantities. It was first suggested by D. K. Jones, et al., in “Optimal Strategies for Measuring Diffusion in Anisotropic Systems by Magnetic Resonance Imaging,” Magnetic Resonance in Medicine, 1999; 42(3):515-525, that the diffusion gradient vectors should be endowed with antipodal symmetry. Because diffusion MRI measurements are acquired sequentially with distinct unit gradient directions, it has been shown that different orderings, or sequences, of the gradient directions have different effects on the quality of tensor-derived quantities obtained from partial scans, or some subset of the complete measurements, as described, for example, by J. Dubois, et al., in “Optimized Diffusion Gradient Orientation Schemes for Corrupted Clinical DTI Data Sets,” Magnetic Resonance Materials in Physics, Biology and Medicine, 2006; 19(3):134-143.
Even though several methods have been proposed to generate optimal orderings of gradient directions, the fact that these methods have not been in routine clinical use may be attributed to two major problems of computational inefficiency. The first problem is that the existing methods for generating highly uniform and antipodally symmetric points on the unit sphere are iterative and inefficient, which may take up to many minutes to several hours to complete, and yet without any clue of whether convergence has been achieved. The second problem is that previously proposed methods for generating optimal ordering of gradient directions are based upon simulated annealing, which takes on the order of 137 hours to generate the ordering for a set of 150 points, as mentioned, for example, by R. Deriche, et al., in “Optimal Real-Time Q-Ball Imaging using Regularized Kalman Filtering with Incremental Orientation Sets,” Medical Image Analysis, 2009; 13(4):564-579.
It would therefore be desirable to provide a method for generating, in a computationally efficient manner, a set of points that are highly uniformly distributed on the surface of a sphere and that exhibit antipodal symmetry.